1. Imperative Languages Section 5 Continued...

2. A Dynamically Typed Language Variable may take on values from different data types. Run­time type checking required. Input and output included. Introduction of ``type tags'' –Storable­value = Tr + Nat –Store = Id  Storable­value –Errvalue = Unit –Expressible­value = –Storable­value + Errvalue

3. Valuation Functions State = Store x Input x Output –store, input and output buffers Post­State = OK + Err –OK = State, Err = State –successful evaluation or type error P: Program  Store  Input  Poststate  –Program takes input store and input buffer and returns a new state

4. C: Command  State  Post­state  –Command takes state and returns a new state. E: Expression  Store  Expressible­value –Expression takes store and returns a value.

5. Composition of States C[[C1 ; C2]] = C[[C1 ]] check­cmd C[[C2 ]] Command = State  Post­state  check­cmd: Command x Command  Command h 1 check­cmd h 2 = a. let z = h1(a) in cases z of isOK(s, i, o)  h 2 (s; i; o) isErr(s, i, o)  z end

6. 1.Give current state a to C[[C 1 ]] producing a post­state z = C[[C 2 ]]a’. 2.If z is a proper state a’ and if the state component is OK, produce C[[C 2 ]]a’ 3.If z is errnoneous, C[[C 2 ]] is ignored and z is the result. Similar for check of expression results.

7. Error Handling Algebra operations abort normal evaluation when type error occurs. –Representation of low­level (e.g. hardware­ level) fault detection and branching mechanisms. Machine action: on fault, branch out of the program. Semantics: on fault, branch out of function expression –Propagation of type errors yields same result as machine action.

8. Altering the Properties of Stores 1. Store critical to evaluation of a phrase. 2. Only one copy of store exists during execution. 3. Store serves as means of communication between phrases. Typical features of a store in sequential programming languages. What happens if we relax each of these restrictions?

9. Delayed Evaluation How to rewrite a function application f(e)? Call­by­value simplification – e evaluated before f is executed – Safe method if f is strict. Call­by­name simplification –f executed with non­evaluated e; –Safe method also if f is non­strict. f: Nat   Nat  f = x: zero f(  ) = zero E[[e]] =  f(E[[e]]) )  ? Simplification of argument may require in­ finite number of steps!

10. Non­strict Store Updates May store operate on improper values? Store = Id  Nat  –Improper values may be stored. update: Id  Nat   Store  Store -- update = i: n: s:[i  n]s -- update[[I]](E[[E]]s) s) is defined even in the``loop forever situation'' E[[E]]s = . -- Unevaluated expressions may be stored in s. E[[E]]s needs not be evaluated until used. -- Delayed (lazy) evaluation. -- Value must be determined with respect to the store that was active when [[E]] was saved.

11. EXAMPLE: begin X := 0 Y := X+1 X := 4 return Y K: Block  Store   Nat  K[[begin C return E]] = s: E[[E]] (C[[C]]s)

12. K[[begin X:=0; Y:= X+1; X:=4 return Y]]s0 = E[[Y]] (C[[X:=0; Y:= X+1; X:=4]]s0 ) = E[[Y]] (C[[Y:= X+1; X:=4]] (C[[X:=0]]s0 )) = E[[Y]] (C[[Y:= X+1; X:=4]] (update [[X]] (E[[0]]s0 ) s0 )) = E[[Y]] (C[[Y:= X+1; X:=4]] s1 ) s 1 = (E[[0]]s0 ) needs not be simplified!

13. EXAMPLE: s2 = update [[Y]] (E[[X+1]]s1 ) s1 s3 = update [[X]] (E[[4]]s 2 ) s2 E[[Y]] (C[[Y:= X+1; X:=4]] s1 ) = E[[Y]]s3 = access [[Y]] s3 = E[[X+1]]s1 = E[[X]]s1 plus one = (access [[X]] s1 ) plus one = E[[0]]s0 plus one = zero plus one = one

14. Non­Strict Command Execution Carry delayed evaluation up to level of commands. Make C, E, and K non­strict in their store arguments. Only those commands need to be evaluated that have an effect on the output of a program!

15. EXAMPLE: begin X:=0; diverge; X:=2 return X+1 K[[begin X:=0; diverge; X:=2 return X+1]]s0 = E[[X+1]] (C[[X:=0; diverge X:=2]]s0 ) = E[[X+1]] (C[[X:=2]](C[[diverge; X:=2]]s0 )) = E[[X+1]](C[[X:=2]]s1 ) = E[[X+1]](update [[X]] (E[[2]]s1 ) s1 ) = E[[X+1]]([ [[X]]  (E[[2]]s1 ) ]s1 ) = E[[X]]([ [[X]]  (E[[2]]s1 ) ]s1 ) plus one = E[[2]]s 1 plus one = two plus one = three s 1 = C[[X:=0; diverge]]s 0 needs not be simplified!

16. Retaining Multiple Stores E[[E 1 + E 2 ]] = s.E[[E 1 ]]s plus E[[E 2 ]]s E uses store s in ``read only'' mode. No second store required for implementation. E[[begin C return E]] = s: let s’ = C[[C]]s in E[[E]]s’ Side effects in expression evaluation possible. Multiple stores required for implementation.

17. E[[E 1 + E 2 ]] = s: let (v’; s’ ) = E[[E 1 ]]s (v’’ ; s’’ ) = E[[E 2 ]]s’ in (v’ plus v’’ ; s’’ ) E[[begin C return E]] = s: let s’ = C[[C]]s in (E[[E]]s’, s’) Local updates remain in global store. Expression evaluation returns new store.

18. Non­Communicating Commands Store facilitates the building up of side effects that lead to some final value. Command advances computation by reading and modifying the values left in the store by previous commands. Otherwise, communication breaks down and language loses sequential flavor.

19. combine: D x D  D Domain s2 Store = Id  D C: Command  Store   Store  C[[C 1;C 2]] = s. join (C[[C1]]s) (C[[C2 ]]s) join: Store   Store   Store  join = s1 : s2:( i:s1 (i) combine s2 (i)) Parallel but non­interfering parallelism.